Page:Elementary Principles in Statistical Mechanics (1902).djvu/201

Rh To make the problem definite, let us consider a system consisting of the original system together with another having the coördinates $$a_1$$, $$a_2$$, etc., and forces $$A_1'$$, $$A_2'$$ etc., tending to increase those coördinates. These are in addition to the forces $$A_1$$, $$A_2$$, etc., exerted by the original system, and are derived from a force-function ($$-\epsilon_q'$$) by the equations For the energy of the whole system we may write  and for the extension-in-phase of the whole system within any limits  or  or again  since $$d\epsilon = dE$$, when $$a_1$$, $$\dot a_1$$, $$a_2$$, $$\dot a_2$$, etc., are constant. If the limits are expressed by $$E$$ and $$E + dE$$, $$a_1$$ and $$a_1 + da_1$$, $$\dot a_1$$ and $$a_1 + d\dot a_1$$, etc., the integral reduces to The values of $$a_1$$, $$\dot a_1$$, $$a_2$$, $$\dot a_2$$, etc., which make this expression a maximum for constant values of the energy of the whole system and of the differentials $$dE$$, $$da_1$$, $$d\dot a_1$$, etc., are what may be called the most probable values of $$a_1$$, $$\dot a_1$$, etc., in an ensemble in which the whole system is distributed microcanonically. To determine these values we have when  That is,